Purity-bounded uncertainty relations in multidimensional space -- generalized purity

Uncertainty relations for mixed quantum states (precisely, purity-bounded position-momentum relations, developed by Bastiaans and then by Man'ko and Dodonov) are studied in general multi-dimensional case. An expression for family of mixed states at the lower bound of uncertainty relation is obtained. It is shown, that in case of entropy-bounded uncertainty relations, lower-bound state is thermal, and a transition from one-dimensional problem to multi-dimensional one is trivial. Results of numerical calculation of the relation lower bound for different types of generalized purity are presented. Analytical expressions for general purity-bounded relations for highly mixed states are obtained.Comment: 12 pages, 2 figures. draft version, to appear in J. Phys. A Partially based on a poster "Multidimensional uncertainty relations for states with given generalized purity" presented on X Intl. Conf. on Quantum Optics'2004 (Minsk, Belarus, May 30 -- June 3, 2004) More actual report is to be presented on ICSSUR-2005, Besan\c{c}on, France and on EQEC'05, Munich. V. 5: amended article after referees' remark

Direct Measurement of Kirkwood-Rihaczek distribution for spatial properties of coherent light beam

We present direct measurement of Kirkwood-Rihaczek (KR) distribution for spatial properties of coherent light beam in terms of position and momentum (angle) coordinates. We employ a two-local oscillator (LO) balanced heterodyne detection (BHD) to simultaneously extract distribution of transverse position and momentum of a light beam. The two-LO BHD could measure KR distribution for any complex wave field (including quantum mechanical wave function) without applying tomography methods (inverse Radon transformation). Transformation of KR distribution to Wigner, Glauber Sudarshan P- and Husimi or Q- distributions in spatial coordinates are illustrated through experimental data. The direct measurement of KR distribution could provide local information of wave field, which is suitable for studying particle properties of a quantum system. While Wigner function is suitable for studying wave properties such as interference, and hence provides nonlocal information of the wave field. The method developed here can be used for exploring spatial quantum state for quantum mapping and computing, optical phase space imaging for biomedical applications.Comment: 27 pages, 14 figure

Harmonic states for the free particle

Different families of states, which are solutions of the time-dependent free Schr\"odinger equation, are imported from the harmonic oscillator using the Quantum Arnold Transformation introduced in a previous paper. Among them, infinite series of states are given that are normalizable, expand the whole space of solutions, are spatially multi-localized and are eigenstates of a suitably defined number operator. Associated with these states new sets of coherent and squeezed states for the free particle are defined representing traveling, squeezed, multi-localized wave packets. These states are also constructed in higher dimensions, leading to the quantum mechanical version of the Hermite-Gauss and Laguerre-Gauss states of paraxial wave optics. Some applications of these new families of states and procedures to experimentally realize and manipulate them are outlined.Comment: 21 pages, 3 figures. Title changed, content added, references adde

Universality of pseudogap and emergent order in lightly doped Mott insulators

It is widely believed that high-temperature superconductivity in the cuprates emerges from doped Mott insulators. The physics of the parent state seems deceivingly simple: The hopping of the electrons from site to site is prohibited because their on-site Coulomb repulsion U is larger than the kinetic energy gain t. When doping these materials by inserting a small percentage of extra carriers, the electrons become mobile but the strong correlations from the Mott state are thought to survive; inhomogeneous electronic order, a mysterious pseudogap and, eventually, superconductivity appear. How the insertion of dopant atoms drives this evolution is not known, nor whether these phenomena are mere distractions specific to hole-doped cuprates or represent the genuine physics of doped Mott insulators. Here, we visualize the evolution of the electronic states of (Sr1-xLax)2IrO4, which is an effective spin-1/2 Mott insulator like the cuprates, but is chemically radically different. Using spectroscopic-imaging STM, we find that for doping concentration of x=5%, an inhomogeneous, phase separated state emerges, with the nucleation of pseudogap puddles around clusters of dopant atoms. Within these puddles, we observe the same glassy electronic order that is so iconic for the underdoped cuprates. Further, we illuminate the genesis of this state using the unique possibility to localize dopant atoms on topographs in these samples. At low doping, we find evidence for much deeper trapping of carriers compared to the cuprates. This leads to fully gapped spectra with the chemical potential at mid-gap, which abruptly collapse at a threshold of around 4%. Our results clarify the melting of the Mott state, and establish phase separation and electronic order as generic features of doped Mott insulators.Comment: This version contains the supplementary information and small updates on figures and tex

Quantum theta functions and Gabor frames for modulation spaces

Representations of the celebrated Heisenberg commutation relations in quantum mechanics and their exponentiated versions form the starting point for a number of basic constructions, both in mathematics and mathematical physics (geometric quantization, quantum tori, classical and quantum theta functions) and signal analysis (Gabor analysis). In this paper we try to bridge the two communities, represented by the two co--authors: that of noncommutative geometry and that of signal analysis. After providing a brief comparative dictionary of the two languages, we will show e.g. that the Janssen representation of Gabor frames with generalized Gaussians as Gabor atoms yields in a natural way quantum theta functions, and that the Rieffel scalar product and associativity relations underlie both the functional equations for quantum thetas and the Fundamental Identity of Gabor analysis.Comment: 38 pages, typos corrected, MSC class change